There have hitherto been known calibration devices for performing camera calibration for photographing devices or projecting devices (e.g., see Patent Literature 1 and Non-Patent Literatures 1 and 2). A camera model includes a plurality of unknown parameters (camera parameters). By obtaining camera parameters in advance by using a calibration device, it is possible to mathematically obtain principal rays in the real world, corresponding to two-dimensional coordinates (pixel coordinates) of an image. The principal rays are also called back-projected straight lines or lines of sight corresponding to pixel coordinates.
Here, the conventional camera calibration disclosed in Patent Literature 1 and Non-Patent Literature 1 will be described. Camera calibration is performed by the following procedure by using a mathematical camera model representing a process of capturing an image of three-dimensional coordinates in the real world with a camera and then transforming the three-dimensional coordinates into two-dimensional coordinates of an image. First, three-dimensional coordinates in the real world (hereinafter referred to as world coordinates) (x, y, z) are projected to normalized image-surface coordinates (up, vp) by using Eq. 1.
                    {                                                                              u                  p                                =                                                                                                    r                        11                                            ⁢                      x                                        +                                                                  r                        12                                            ⁢                      y                                        +                                                                  r                        13                                            ⁢                      z                                        +                                          t                      x                                                                                                                          r                        31                                            ⁢                      x                                        +                                                                  r                        32                                            ⁢                      y                                        +                                                                  r                        33                                            ⁢                      z                                        +                                          t                      z                                                                                                                                                                v                  p                                =                                                                                                    r                        21                                            ⁢                      x                                        +                                                                  r                        22                                            ⁢                      y                                        +                                                                  r                        23                                            ⁢                      z                                        +                                          t                      y                                                                                                                          r                        31                                            ⁢                      x                                        +                                                                  r                        32                                            ⁢                      y                                        +                                                                  r                        33                                            ⁢                      z                                        +                                          t                      z                                                                                                                              {                  Eq          .                                          ⁢          1                }                                          R          =                      (                                                                                r                    11                                                                                        r                    12                                                                                        r                    13                                                                                                                    r                    21                                                                                        r                    22                                                                                        r                    23                                                                                                                    r                    31                                                                                        r                    32                                                                                        r                    33                                                                        )                          ,                  T          =                      (                                                                                t                    x                                                                                                                    t                    y                                                                                                                    t                    z                                                                        )                                              {                  Eq          .                                          ⁢          2                }            
Note that a rotation matrix R and a translation vector T in Eq. 2 represent three-dimensional coordinate transformation from the world coordinates to the camera coordinates. These are values representing the position and orientation of the camera in relation to the world coordinates and are referred to as external parameters.
Note that Eq. 1 is based on the assumption that all principal rays intersect at the optical center of the camera. Then, (ud, vd), which is the normalized image-surface coordinates (up, vp) with distortion aberration added thereto, is obtained.
                    {                                                                              u                  d                                =                                                      u                    p                                    +                                                            g                      1                                        ⁡                                          (                                                                        u                          p                          2                                                +                                                  v                          p                          2                                                                    )                                                        +                                                            g                      3                                        ⁢                                          u                      p                      2                                                        +                                                            g                      4                                        ⁢                                          u                      p                                        ⁢                                          v                      p                                                        +                                                            k                      1                                        ⁢                                                                  u                        p                                            ⁡                                              (                                                                              u                            p                            2                                                    +                                                      v                            p                            2                                                                          )                                                                                                                                                                                      v                  d                                =                                                      v                    p                                    +                                                            g                      2                                        ⁡                                          (                                                                        u                          p                          2                                                +                                                  v                          p                          2                                                                    )                                                        +                                                            g                      3                                        ⁢                                          u                      p                                        ⁢                                          v                      p                                                        +                                                            g                      4                                        ⁢                                          v                      p                      2                                                        +                                                            k                      1                                        ⁢                                                                  v                        p                                            ⁡                                              (                                                                              u                            p                            2                                                    +                                                      v                            p                            2                                                                          )                                                                                                                                                    {                  Eq          .                                          ⁢          3                }            
Here, (g1, g2, g3, g4, k1) are distortion parameters. Furthermore, by using Eq. 4, (ud, vd), which is the normalized image-surface coordinates with the distortion aberration added thereto, is transformed into per-pixel pixel coordinates (u, v).
                    {                                                            u                =                                                                            α                      u                                        ⁢                                          u                      d                                                        +                                      u                    0                                                                                                                          v                =                                                                            α                      v                                        ⁢                                          v                      d                                                        +                                      v                    0                                                                                                          {                  Eq          .                                          ⁢          4                }            
In the standard camera model, transformation from the world coordinates (x, y, z) to the pixel coordinates (u, v), associated with image capturing by means of a camera, is expressed by Eqs. 1 to 4, as described above. Note that the parameters (αu, αv, u0, v0, g1, g2, g3, g4, k1) in Eqs. 3 and 4 represent the characteristics of the camera itself and are thus referred to as internal parameters.
Distortion parameters are defined in various ways depending on the application. For example, Eq. 3 is a model in which distortion aberrations of not higher than the third order are taken into consideration; however, models in which higher-order terms are further added, like the fifth order, the seventh order, and so forth, are also used. The representative distortion model among these models is the Brown model of Non-Patent Literature 2, expressed in Eq. 5.
                                          (                                                                                u                    d                                                                                                                    v                    d                                                                        )                    =                                    (                                                                                          u                      p                                                                                                                                  v                      p                                                                                  )                        +                                          (                                                                            k                      1                                        ⁢                                          r                      p                      2                                                        +                                                            k                      2                                        ⁢                                          r                      p                      4                                                        +                                                            k                      3                                        ⁢                                          r                      p                      6                                                        +                  …                                ⁢                                                                  )                            ⁢                                                          ⁢                              (                                                                                                    u                        p                                                                                                                                                v                        p                                                                                            )                                      +                                          [                                                                  ⁢                                                                            p                      1                                        ⁡                                          (                                                                                                                                                                  ρ                                p                                2                                                            +                                                              2                                ⁢                                                                  u                                  p                                  2                                                                                                                                                                                                                                                        2                              ⁢                                                              u                                p                                                            ⁢                                                              v                                p                                                                                                                                                        )                                                        +                                                            p                      2                                        ⁡                                          (                                                                                                                                  2                              ⁢                                                              u                                p                                                            ⁢                                                              v                                p                                                                                                                                                                                                                                                        ρ                                p                                2                                                            +                                                              2                                ⁢                                                                  v                                  p                                  2                                                                                                                                                                                        )                                                                      ]                            ⁢                              (                                  1                  +                                                            p                      3                                        ⁢                                          ρ                      p                      2                                                        +                  …                                ⁢                                                                  )                                                    ⁢                                  ⁢                                  ⁢                              where            ⁢                                                  ⁢                          ρ              p              2                                =                                    u              p              2                        +                          v              p              2                                                          {                  Eq          .                                          ⁢          5                }            
In the Brown model, distortion aberrations are expressed by using parameters (k1, k2, k3, . . . ) of rotationally symmetric movement-radius distortion and parameters (p1, p2, p3, . . . ) of rotationally asymmetric tangential distortion.
Generally, in camera calibration, an image of a calibration chart with which the world coordinates (x, y, z) include a plurality of known feature points is captured by means of a camera. Then, pixel coordinates (u, v) at which the feature points were captured in the image are obtained through image processing. Camera parameters are obtained by obtaining a plurality of measurement data representing the correspondence between world coordinates (x, y, z) and pixel coordinates (u, v) in this manner.